Chebyshev series ($f(x)={\sum }_{j\ge 0}{c}_j{T}_j(x)$)

A Chebyshev series expands a function in Chebyshev polynomials $T_j$. If $|c_j|\le M{q}^j$ for $0<q<1$, then $|T_j(x)|\le 1$ on $[-1,1]$ gives uniform convergence and the tail bound $\Vert f-{\sum }_{j=0}^d{c}_j{T}_j{\Vert }_{\infty }\le M{q}^{d+1}/(1-q)$ by the geometric series.